The invention relates to a method for evaluating measured values generated by a sensor which makes polar measurements, and for evaluating the state of an object based thereon.
In many monitoring applications, it is necessary or useful to determine the kinematic state (that is, the position, velocity, acceleration) of an object, by the use of suitable sensors. The measurements which are provided by the respective sensors, however, are always subject to errors. In this regard, a distinction should be made between systematic error (bias) with regard to the measurement variables and a statistical error (a random discrepancy in the measurements from the measured value to be expected on average). It is well known that systematic measurement errors can be suppressed by calibration of the sensors.
One conventional method for estimating the state of an object is to record measurements by means of a sensor repeatedly in a time sequence and to accumulate this information (including the information relating to any statistical measurement uncertainty of the sensor) in conjunction with an assumption about a possible movement behavior of the object. In this way, a statement that is as accurate as possible, relating to the instantaneous state of the object, is obtained from the available information. The mean square error (that is, the average square of the discrepancy to be expected in the estimate from the true state) is generally used as a measure for the reliability of the estimate obtained in the course of this process (which is referred to as tracking). If the estimate has no systematic error (that is, it is bias-free), then the mean square error matches the estimated error variance, that is, the mean square discrepancy between the estimate and the expected value on average. (The square root of the variance, the standard deviation, is often also used instead of the variance). Otherwise, the mean square error is obtained as the sum of the estimated error variance and the square of the bias value. It is frequently also necessary to determine the addressed measures for the reliability of the estimate. However, due to the nature of the problem, this can be done only approximately since the true state of the object is in fact not known during the course of the estimation process.
With the measurements, both the estimated values for the state and the variance (which is in turn estimated) are also subject to unavoidable statistical errors. However, depending on the method that is used, a systematic error can also occur in the estimate of the state (and this despite the assumed freedom from bias in the measurements produced by the sensor). It is therefore possible, for example, for the sensor always to overestimate the range of an object as being too great, with the resulting difference between the range estimated on average and the actual range depending not only on the quality of the sensor (that is, the quality of the measurements), but also on the range (which can be determined only by the estimation method) of the object from the sensor.
Furthermore, estimation methods can be assessed as critical in which the variance as estimated by the system differs significantly from the actual mean square error. This is particularly true in the case of so-called inconsistent estimation methods (that is, when the estimated variance is considerably too small in comparison).
Tracking is therefore frequently used for the purposes of complex technical systems, to track a plurality of objects simultaneously. In this case, one core task is to associate the individual measurements with the respective objects (or to recognize that a particular measurement may be simply an incorrect measurement that does not relate to any of the objects of interest). If an inconsistent method is used in this context, and a measurement which is actually associated with the object cannot be associated with it because the estimate is assumed to be too accurate, then this generally leads to a breakdown in tracking. That is, the system cannot continuously track the object further, and therefore operates incorrectly.
Conventional sensors are used here to carry out polar measurements. That is, the measurement data they produce is expressed in terms of the range from the sensor rm and the azimuth am (angle between north and the horizontal direction to the target, measured in the clockwise sense) Conventional estimation methods used here are intended to estimate the true Cartesian variables x=r sin α and y=r cos α (based on the true distance r and the true azimuth α). In this case, it is assumed that the normally distributed measurement errors are Δrm=rm−r and Δαm=αm−α, which have variances σ2r and α2α.
Proposals for converting polar coordinates to Cartesian pseudo measurements are known from the prior art, including for example from Longbin, Xiaoquan, Yizu, Kang, Bar-Shalom: Unbiased converted measurements for tracking. IEEE Transactions on Aerospace and Electronic Systems vol. 34(3), July 1998, pages 1023-1027 or Miller, Drummond: Comparison of methodologies for mitigating coordinate transformation bias in target tracking. Proceedings SPIE Conference on Signal and Data Processing of Small Targets 2000, vol. 4048, July 2002, pages 414-426 or Duan, Han, Rong Li: Comments on “Unbiased converted measurements for tracking”. IEEE Transactions on Aerospace and Electronic Systems, vol. 40(4), October 2004, pages 1374-1377. In these known techniques, the measurement variables are used to generate pseudo-measurements in the form
                              z          m                =                              [                                                                                x                    m                                                                                                                    y                    m                                                                        ]                    =                      β            ⁢                                                  ⁢                                          r                m                            ⁡                              [                                                                                                    sin                        ⁢                                                                                                  ⁢                                                  α                          m                                                                                                                                                                        cos                        ⁢                                                                                                  ⁢                                                  α                          m                                                                                                                    ]                                                                        (        1.1        )                        and                                                                R          m                =                                            [                                                                                          sin                      ⁢                                                                                          ⁢                                              α                        m                                                                                                                                                                          -                          cos                                                ⁢                                                                                                  ⁢                                                  α                          m                                                                    ⁢                                                                                                                                                                                                      cos                      ⁢                                                                                          ⁢                                              α                        m                                                                                                                        sin                      ⁢                                                                                          ⁢                                              α                        m                                                                                                        ]                        ⁡                          [                                                                                          R                      m                      2                                                                            0                                                                                        0                                                                              C                      m                      2                                                                                  ]                                ⁡                      [                                                                                sin                    ⁢                                                                                  ⁢                                          α                      m                                                                                                            cos                    ⁢                                                                                  ⁢                                          α                      m                                                                                                                                                              -                      cos                                        ⁢                                                                                  ⁢                                          α                      m                                                                                                            sin                    ⁢                                                                                  ⁢                                          α                      m                                                                                            ]                                              (        1.2        )            zm being a Cartesian position measurement with an associated measurement-error variance matrix Rm. In these expressions, R2m and C2m are the variables (each dependent on rm, but not on αm) which the method assumes as the nominal equivalent measurement-error variance in the direction of the target (the variance R2m in the range direction) and transversely thereto (the variance C2m). Herein, R2m is also referred to as the variance in the range direction and C2m is referred to as the variance in the crossrange direction.
According to equation (1.1), in the known techniques the Cartesian (pseudo-) position measurement zm is obtained by conventional conversion from polar to Cartesian coordinates followed by a multiplication correction, with the scaling factor β being intended to correct a bias which would result from the estimation process in the case where β=1. The Cartesian (pseudo-) measurement-error variance Rm according to equation (1.2) defines an ellipse (over zTRmz=constant), of which one of the mutually perpendicular major axes is aligned with the measurement direction, in the same way as zm (the assumed Cartesian covariance matrix therefore rotated through the measured angle αm with respect to a diagonal matrix having the major diagonal elements R2m and C2m). The use of the variables zm and Rm as a Cartesian pseudo-measurement in this case expresses the fact, for example, that a Kalman filter is used as the estimator, in an updating of the position estimate according to:S=Pp+Rm,K=PpS−1,zu=zp+K(zm−zp),Pu=Pp−KSKT  (1.3)for estimates zp (for the position) and Pp (for the associated estimated error variance) before and corresponding to zu and Pu, after consideration of the measurement.
FIG. 1 illustrates the described procedure. The figure indicates the position of a stationary target 1 at a range r=10 km and an azimuth α=0°. The solid line in FIG. 1.a is the 90% confidence region of the measurements (that is, the region in which, on average, 90% of all measurements can be expected) with a standard deviation of σr=50 m for the range measurement and σα, =15° for the azimuth measurement. The illustration shows three measurements 2 with different range and azimuth errors in the form of the associated uncorrected (that is, obtained using β=1) Cartesian pseudo-measurements.
FIG. 1.b shows the situation in detail, illustrating, in addition to the pseudo-measurements 2 i) the 90% confidence ellipses 2a which are respectively associated with them and are specified by Rm (that is, in each case that region in which there is a 90% probability of the targets supposedly being located, dashed lines), ii) the estimate 3 obtained and iii) the 90% confidence ellipse 3a which is associated with this estimate by the method.
The advantages of a procedure such as this over other known methods for use in technical systems for tracking a plurality of objects include i) a simple updating step, and ii) a calculation of the Cartesian pseudo-measurements requiring no knowledge whatsoever about the (estimated) state of the object. Therefore it need be carried out only once per measurement (and not, for example, once per combination of object/measurement or even object/motion model/measurement) Overall, methods such as these are therefore relatively less computation intensive, and are therefore particularly suitable for use in real-time systems.
The following are known as prior art from the literature as variants of methods which use Cartesian pseudo-measurements for the purposes of an estimation process for sensors which produce polar measurements:
Method 1 (classical):β=1R2m=σ2r C2m=r2mσ2α  (1.4)
Method 2 (Longbin et.al. 1998):β=1R2rn=λ2(cos h(σ2α)−1)(r2m+σ2r)+σ2r)2(cos h(σ2α)−1)r2m C2m=λ2 sin h(σ2α)(r2m+σ2r)  (1.5)
Method 3 (Miller & Drummond 2002, Duan et. al. 2004):β=λR2m=λ2((cos h(σ2α)−1)(r2m+σ2r)+σ2r)C2m=λ2 sin h(σ2α)(r2m+σ2r)  (1.6)whereλ=exp(−σ2α/2)  (1.7)
These methods all share the common disadvantage, however, that they lead to a systematic error (bias) in the range direction for objects which represent stationary (non-moving) targets and for targets which are not maneuvering or are scarcely maneuvering (that is, which move at an exactly or approximately constant velocity). Furthermore, none of these methods produces consistent estimates for the variance in the crossrange direction. Each of the above methods has the weaknesses mentioned above.
One object of the present invention, therefore, is to provide a method which overcomes the disadvantages of the prior art.
This and other objects and advantages are achieved by the method according to the invention, which comprises the following steps:                a sensor makes a number n>1 of polar measurements with respect to an object that is to be detected;        the polar measurements are converted to Cartesian pseudo-measurements zm by conversion of the polar measured values rm and αm to Cartesian coordinates and with subsequent scaling by means of a scaling factor β which is calculated suitably as a function of the measured range rm;        associated pseudo-measurement error variance matrices are determined, each suitably comprising specific nominal measurement-error variances R2m in the range direction and C2m transversely thereto, as a function of the measured range rm;        based on the Cartesian pseudo-measurements and the pseudo-measurement error variance matrices, an estimation device makes an estimate of the state of the object, with an estimated variance {circumflex over (σ)}2cross being determined transversely with respect to the range direction.        
In this case, the method according to the invention is distinguished by the following advantageous special features:                the scaling factor β is chosen such that no systematic error results for a position estimate comprising n>1 measurements; and        the nominal pseudo-measurement error variance R2m in the range direction is calculated as a function of the nominal pseudo-measurement error variance C2m, transversely thereto, or conversely such that the variance {circumflex over (σ)}2cross which is estimated after the processing of n>1 measurements, transversely with respect to the range direction, on average matches the actual variance {circumflex over (σ)}2cross (as is to be expected after the processing of these n>1 measurements) of the estimated error transversely with respect to the range direction.        
Other objects, advantages and novel features of the present invention will become apparent from the following detailed description of the invention when considered in conjunction with the accompanying drawings.